Entropy stable numerical approximations for the isothermal and polytropic Euler equations

摘要: In this work we analyze the entropic properties of Euler equations when system is closed with assumption a polytropic gas. case, pressure solely depends upon density fluid and energy equation not necessary anymore as mass conservation momentum then form system. Further, total acts convex mathematical entropy function for equations. The state gives scaled power law in terms adiabatic index $$\gamma $$ . As such, there are important limiting cases contained within model like isothermal ( $$\gamma {=}1$$ ) shallow water {=}2$$ ). We first mimic continuous analysis on discrete level finite volume context to get special numerical flux functions. Next, these fluxes incorporated into particular discontinuous Galerkin (DG) spectral element framework where derivatives approximated summation-by-parts operators. This guarantees high-order accurate DG approximation that also consistent its auxiliary behavior. Numerical examples provided verify theoretical derivations, i.e., high order scheme.